The lengths of the legs of an isosceles triangle are integers. The base is half as long as each leg. What are the possible lengths of the legs if the perimeter is between 6 units and 16 units?

Question

The lengths of the legs of an isosceles triangle are integers.  The base is half as long as each leg.  What are the possible lengths of the legs if the perimeter is between 6 units and 16 units?

anonymous · Answer

If a is the length ah a leg and b is the length of the base, then b = a/2 and the perimeter is 5a/2.  Since 6 < 5a/2 < 16, 12/5 < a < 32/5.  Because a is an integer, a can be 3, 4, 5, or 6.

anonymous · Answer

Use abtrehearn's notation, p = 5a/2.  If p is ranged from 6 through 16 we are interested in the interger "a" values.  The following Mathematica method shows that only two "a" values are integers, 4 and 6, when p is ranged from 6 through 16.

$$\text{Table}[\text{Solve}[a 5/2==i,a],\{i,6,16\}]= $$$$\left\{\frac{12}{5},\frac{14}{5},\frac{16}{5},\frac{18}{5},4,\frac{22}{5},\frac{24}{5},\frac{26}{5},\frac{28}{5},6,\frac{32}{5}\right\}  $$